You have two issues:
First, you are using document.write() which should not be used as it will cause any previously loaded document to be thrown out and a new one created every time you use it. document.write() is a very old API that has very limited uses in modern JavaScript code. Instead, you should write the value to a pre-existing element that is already on the page.
Second, all data in HTML is strings. To use them as numbers in JavaScript, you have to convert them (either implicitly or explicitly). There are several ways to do this, but one way is to prepend the string with +.
<html>
<head>
<title></title>
</head>
<body>
<!-- The output from the script will be placed
within this element. -->
<div></div>
<!-- The default type for a script is JavaScript, so you don't
need to write that. -->
<script>
// Get a reference to the div
const output = document.querySelector("div");
// window is the global object in a browser and
// so you can omit explicitly stating it
var minNum = +prompt("Enter min number:");
var maxNum = +prompt("Enter max number:");
if (maxNum - minNum != 1 && maxNum > minNum) {
for (i = (minNum + 1); i < maxNum; i++){
// To be able to have the HTML portion parsed
// as HTML, use .innerHTML on the output element.
// Also, to make the new value be appended to
// the current value, we use += instead of =.
output.innerHTML += i + "<br>";
}
}
</script>
</body>
</html>
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